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Hypergeometric series and irrationality of the values of the Riemann zeta function. (Séries hypergéométriques et irrationalité des valeurs de la fonction zêta de Riemann.) (French) Zbl 1041.11051

The author presents a survey of the ideas around his striking discoveries about the values of the Riemann zeta-function at the odd integers. He shows how hypergeometric series and Padé approximants provide a common approach to Apéry’s theorem on the irrationality of \(\zeta(3)\), the author’s remarkable proof that infinitely many of the \(\zeta(2n+1)\) are irrational, and Zudilin’s theorem, improving another result of the author, that at least one of the numbers \(\zeta(5),\zeta(7),\zeta(9),\zeta(11)\) is irrational. Some teasing questions are added in the conclusion.

MSC:

11J72 Irrationality; linear independence over a field
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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References:

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